The Control of Linear Systems under Feedback Delays A. N. Daryin, - - PowerPoint PPT Presentation

The Control of Linear Systems under Feedback Delays

• A. N. Daryin, A. B. Kurzhanski, and I. V. Vostrikov

Moscow State (Lomonosov) University Faculty of Computational Mathematics and Cybernetics

Vienna Conference on Mathematical Modelling, 2009

Introduction

The emphasis in this paper is on Feedback delays Measurement feedback Set-membership noise Effect of delay Systems of high dimensions The solution is based on Hamiltonian techniques Convex analysis Ellipsoidal calculus Numerical modelling Effect of delay Oscillating systems High dimensions (here up to 20)

Introduction

Problems:

1 Feedback delay, no noise 2 Noisy measurement feedback, no delay 3 Noisy measurement feedback + delay

Problem 1 - Feedback delay, no noise

System: ˙ x(t) = A(t)x(t) + B(t)u

• n fixed time interval t ∈ [t0, t1].

Hard bound: u[t] ∈ P(t) ∈ conv Rn Feedback control: u = u[t] = U(t, x(t − h)). The controller is allowed to be with memory. Target control: x(t1) ∈ M .

Problem 1 - Feedback delay, no noise

Solution: reduce to system without delay. For t ∈ [t0, t0 + h): set u = 0. For t t0 + h: consider system ˙ z(t) = A(t)z(t) + B(t)u[t], z(t0 + h) = X(t0 + h, t0)x(t0). (here ∂X(t, τ)/∂t = A(t)X(t, τ), X(τ, τ) = I). State z(t) is available without delay ⇒ construct feedback control u = U(t, z). Reset at time τ: z(τ) := X(τ, τ − h)x(τ − h) + zτ(τ). ˙ zτ(t) = A(t)zτ(t) + B(t)u[t], zτ(τ − h) = 0, t ∈ [τ − h, τ]

Problem 2 - Measurement feedback, no delay

Measurement equation: y(t) = H(t)x(t) + ξ(t) Hard bound: ξ[t] ∈ Q(t) ∈ conv Rn Feedback control: u = u[t] = U(t, y(t)) with memory.

Problem 2 - Measurement feedback, no delay

Solution: Information set X [τ] (guaranteed state estimation) lim

σ→0+0 σ−1h(X (t+σ), (X (t)+σB(t)u∗[t])∩(y ∗(t)−Q(t))) = 0.

State {τ, X [τ]} ⇒ infinite-dimension problem (metric space

• f convex compacts).

But: it reduces to a finite-dimension problem through techniques of convex analysis (see paper for details).

Problem 3 - Measurement feedback + delay

Measurement equation: y(t − h) = H(t − h)x(t − h) + ξ(t − h), ξ(t − h) ∈ Q(t − h). Time delay h. Feedback control: u = u[t] = U(t, y(t − h)) with memory. Solution: combine techniques for Problems 1 and 2 State: {t, X (t − h), u[t − h, t]} Notation: Tuu[τ, t] = t

τ

X(t, ϑ)B(ϑ)u(ϑ)dϑ. Guaranteed estimate of the current position x(t): X ∗[t] = X(t, t − h)X (t − h) + Tuu[t − h, t].

Problem 3 - Measurement feedback + delay

Notation: Geometric (Minkowski) difference: A ˙ − B = {x ∈ Rn | x + B ⊆ A}

A B A − B .

Problem 3 - Measurement feedback + delay

Solution (cont.): Estimate of the value function: V (t, X (t − h), u[t − h, t]) d(X(t1, t)Tuu[t − h, t], M ˙ − X(t1, t − h)X (t − h) − TuP[t, t1]) It is equal to value function of a linear-convex problem with:

Target set M ˙ − X(t1, t − h)X (t − h) Initial position x(t) = Tuu[t − h, t] No delay No noise

Apply Ellipsoidal Calculus to this problem.

Examples

k1 k2 m1 m2 w1 w2 kN mN−1 mN wN−1 wN u Only positions wi measured Control applied to lower weight Completely controllable Completely observable

N = 1: Information Tube

0.05 0.1 −0.1 −0.05 0.05 0.1 −2 −1 1 2 Time w1 w’1

Size of the Information Tube

10

−2

10

−1

10 10

−2

10

−1

10 10

1

10

2

Time From Start (t − t0) Size of Ellipsoidal Estimation

N = 2

N = 2: Effect of Delay

10 20 30 40 5 10 15 20 25 30 35 40 Feedback Delay (h) Final Energy (E)

N = 10

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